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Physics-Informed Neural Networks for Fractional Integro-Differential-Algebraic Equations: A Mesh-Free Framework with Convergence Analysis |
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PP: 585-607 |
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doi:10.18576/pfda/120307
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Author(s) |
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Amin M. Elfeky,
Adel S. Hussain,
Essa Melfi Aldosari,
Mohammad A. Tashtoush,
Mohamed Sayed Abdellatif,
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Abstract |
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| Fractional integro-differential-algebraic equations (FIDAEs) provide a mathematical framework for describing dynamical systems characterized by nonlocal memory effects, hereditary behavior, and algebraic constraints. However, their numerical approximation remains challenging because of the coupling between fractional differential operators, integral terms, and algebraic conditions. This study presents a modified physics-informed neural network (PINN) framework for approximating solutions to FIDAEs, including fractional Bagley–Torvik-type models. The proposed approach incorporates fractional differential operators, Volterra-type integral terms, initial conditions, and algebraic constraints into a unified physics-informed loss function. Numerical quadrature is employed to approximate the nonlocal operators, while neural network optimization is used to minimize the governing equation residuals. The mathematical formulation is accompanied by a convergence analysis that discusses the approximation properties of the proposed framework under regularity and continuity assumptions. The computational performance is investigated through representative fractional integro-differential problems with available reference solutions. Numerical comparisons based on pointwise absolute errors demonstrate that the proposed method can produce accurate approximations and competitive results relative to previously reported numerical approaches. The findings illustrate the potential of physics-informed neural networks as a flexible computational tool for fractional dynamical systems with nonlocal operators and algebraic constraints, while identifying the need for further theoretical and computational investigation of convergence and stability. |
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